An abstract model for aggregated connectionless traffic, based on the fractional Brownian motion, is presented. Insight into the parameters is obtained by relating the model to an equivalent burst model. Results on a corresponding storage process are presented. The buffer occupancy distribution is approximated by a Weibull distribution. The model is compared with publicly available samples of real Ethernet traffic. The degree of the short-term predictability of the traffic model is clarified through an exact formula for the conditional variance of a future value given the past. The applicability and interpretation of the self-similar model are discussed extensively, and the notion of ideal Free Traffic is introduced. Keywords: LAN traffic, long-range dependence, self-similar, prediction 1 Introduction In this paper we are considering the modelling of traffical phenomena in a connectionless network. The principle of such a network is that all data is sent in relatively small independen...

A Wavelet based tool for the analysis of long range dependence is introduced and a related semiparametric estimator of the Hurst parameter. The estimator is shown to be unbiased under very general conditions, and efficient under Gaussian assumptions. It can be implemented very efficiently allowing the direct analysis of very large data sets, and is highly robust against the presence of deterministic trends, as well as allowing their detection and identification. Statistical, computational and numerical comparisons are made against traditional estimators including that of Whittle. The estimator is used to perform a thorough analysis of the long range dependence in Ethernet traffic traces. New features are found with important implications for the choice of valid models for performance evaluation. A study of mono vs multi-fractality is also performed, and a preliminary study of the stationarity with respect to the Hurst parameter and deterministic trends.

A wavelet-based tool for the analysis of long-range dependence and a related semi-parametric estimator of the Hurst parameter is introduced. The estimator is shown to be unbiased under very general conditions, and efficient under Gaussian assumptions. It can be implemented very efficiently allowing the direct analysis of very large data sets, and is highly robust against the presence of deterministic trends, as well as allowing their detection and identification. Statistical, computational, and numerical comparisons are made against traditional estimators including that of Whittle. The estimator is used to perform a thorough analysis of the long-range dependence in Ethernet traffic traces. New features are found with important implications for the choice of valid models for performance evaluation. A study of mono versus multifractality is also performed, and a preliminary study of the stationarity with respect to the Hurst parameter and deterministic trends.

As ATM high-speed, cell-relay networks will most likely first make their impact as backbones interconnecting enterprise networks consisting of Ethernet and other LANs, their proper design and control is crucial. Recent studies of high quality, high resolution traffic measurements in Bellcore Ethernets have revealed that this aggregate Ethernet traffic is self-similar ("fractal ") in nature, quite different in "burstiness" features from traffic considered and studied up to now. This paper presents an analytical study of an ATM buffer driven with self-similar traffic. The probability of buffer occupancy is obtained. It is shown that this probability decreases with the buffer size not exponentially, as in traditionally Markovian traffic models, but algebraically . 1 Introduction Recent studies of high-quality, high resolution traffic measurements have revealed a new phenomenon with potentially important ramifications to the modeling, design and control of broadband networks. These includ...

In previous work [24], Fractal Point Processes (FPPs) have been proposed as novel tools for understanding, modeling and analyzing diverse types of self-similar traffic behavior. We apply the FPP models in the context of network traffic modeling and performance analysis. Two qualitatively different fractal data sets (Bellcore Ethernet traces) are characterized by FPP models. Comparison of model-driven and trace-driven queueing simulation results show that the matched models yield close agreement with the traces over a wide range of system parameters. We also show that under suitable conditions, the FPP models yield Gaussian processes. Queueing simulation shows that the FPP models can be computationally efficient alternatives for generating fractional Gaussian noise processes. Finally, we divide fractal traffic into two types, applicationlevel fractal traffic and network-level fractal traffic, and argue that each type has radically different implications for the design and control of fut...

this paper, we analyze a fluid or storage queueing system with LRD input. Fluid systems have been used before (e.g. Bensaou et al. [2], Guibert [7]) to model bursty traffic fed into ATM multiplexer queues, when considering time scales where the granularity of the ATM cells no longer dominates. The input sources are assumed to be of On/Off type, that is, with mutually independent, alternating silence periods (no work arriving) and activity periods (work arriving at a constant rate). We consider a superposition of N identical, independent On/Off sources flowing into an infinite reservoir with fixed output rate C. The object of study is the complementary distribution function Q of the stationary queue content. F. Brichet et al. / Heavy traffic analysis with LRD sources 3 If is the mean arrival rate for a single source, we require C ? N for stability. For N sufficiently large, C then exceeds the peak rate of an individual source. If this rate is proportional to C=N , so that it decreases with N , then we are in the realm of "small" sources. The M=G=1 example above represents the limiting form of the alternative assumption, where instantaneous arrivals idealize the case of a capacity significantly smaller than the individual arrival rate. By selecting "heavy" tails (as defined in section 4) for the silence and/or activity periods, the input process becomes long range dependent and the queueing problem is fundamentally non-Markovian. Despite this, a useful lower bound L to

Self-similar processes based on fractal point processes (FPPs) provide natural and attractive network tra#c models. We show that the point process formulation yields a wide range of FPPs which in turn yield a diversity of parsimonious, computationally e#cient, and highly practical asymptotic second-order self-similar processes. Using this framework, we show that the relevant second-order fractal characteristics such as long-range dependence (LRD), slowly-decaying variance, and 1/f noise are completely characterized by three fundamental quantities: mean arrival rate, Hurst parameter, and fractal onset time. Four models are proposed, and the relationship between their model parameters and the three fundamental quantities are analyzed. By successfully applying the proposed models to Bellcore's Ethernet traces, we show that the FPP models prove useful in evaluating and predicting the queueing performance of various types of fractal tra#c sources. Keywords: point process, fractal, self-similarity, long-range dependence, tra#c modeling 1 Throughout this paper, self-similarity refers to asymptotic second-order self-similarity [4], [13] unless otherwise defined. 1

It has now been demonstrated in many studies that network traffic exhibits properties consistent with Long Range Dependence (LRD) and self-similarity. While theoretical frameworks are currently being developed to estimate the performance of such systems, simulation will remain a valuable tool for validating these theoretical models, and providing insight into systems which are too complicated to effectively model. Furthermore, when testing real systems, it is desirable to have traffic sources which are realistic, and hence display self-similarity. The Fractal Renewal Process (FRP) and its variants (including On/Off processes and superpositions thereof) have been proposed as models for LRD processes, in particular for network traffic. The FRP is a simple renewal point process with heavy-tailed inter-renewal times. The long-range correlations in the process are directly introduced by the heavy tail of the renewal times. The FRP has the great advantage that the number of computations required to generate a time series is linear and the time series can be generated on-line, facilitating generation of real traffic. However, there are some problems which arise when using such processes to generate LRD traffic. Most notably undersampling of the heavy-tailed random variables used to generate FRPs can lead to a truncation of the sampled autocorrelation that is not consistent with LRD. This problem becomes clear when the processes are investigated using the wavelet based methods of Abry and Veitch which segregate behaviour at different scales. This paper will describe the problem of undersampling, and its effects, and methods for avoiding the problem.

We investigate the application of deterministic chaotic maps to model traffic sources in packet based networks, motivated in part by recent measurement studies which indicate the presence of significant statistical features in packet traffic more characteristic of fractal processes than conventional stochastic processes. We outline one approach whereby traffic sources can be modeled by chaotic maps, and illustrate the traffic characteristics that can be generated by analyzing three maps. We show that low order nonlinear maps can capture several of the fractal properties observed in actual data. Finally, we outline a potential performance analysis approach based on chaotic maps that can be used to assess the traffic significance of fractal properties. It is our conclusion that while there are considerable analytical difficulties, chaotic maps may allow accurate, yet concise, models of packet traffic, with some potential for transient and steady state analysis.

We consider a fluid queueing problem with an infinite reservoir with output rate C, fed by an infinite superposition of identical On/Off sources with large instantaneous bit rate h C. The On periods have infinite variance, generating Long Range Dependence. It is shown that the distribution of the stationary queue content at embedded time points is heavy tailed with infinite mean. It is explained why this tail behaviour is found rather than Weibullian behaviour which can also arise with heavy On/Off sources. The equilibrium distribution of the maximum queue content during a busy period is shown to be heavy tailed with infinite variance. Keywords: On/Off Fluid Sources, Heavy Traffic, Long Range Dependence, Power law queues. 1 Introduction The classical queueing literature is concerned essentially with systems with exponentially decaying covariance (short term dependence) in their arrival processes. This gives rise to distributions of queue content or virtual waiting time with tails wh...